A question concerning Evolution

are NOT the same kind of mathematical object.People always seem to get this fact wrong.A black hole solution in GR is embedded in an exterior spacetime. You need an exterior spacetime to define the concept of event horizon.A Big Bang singularity has NO exterior spacetime for an event horizon to be defined in. Thus it is a source for all particle worldlines which, of course, does not apply to a black (or white) hole solution where worldlines don't have to intersect the singularity.

The expansion of spacetime is a global cosmological solution to a FLRW metric. It is not local.Locally we solve GR by say the Schwarzchild solution. There is no expansion of spacetime here.So the common answer is that your two objects are not expanding away from each other.Cop out answer, right?In reality the metric of the Universe cannot be approximated by these well known analytic solutions.So there is probably an expansion of spacetime in the background so to speak but the effect is negligible. Sort of like the Earth's orbit shifting an atomic width over the lifetime of the solar system.If we had a situation like your example 2) then the energy would have to come from whatever field was powering the exapnsion itself.This brings me roundabout to a comment or two on the 'dark energy'.What is it?Is it a cosmological constant related to the vacuum energy density - maybe but the math doesn't add up by current theory.Is it not constant but some new force a la quintessence models?Is this quintessence a decaying field left over from inflation? Who knows.Probably the biggest mystery in physics today is the nature of the dark energy. We have so little to go on - and I doubt it is solved soon.

I knew someone addressed issues like this in a paper I read once. I just found it. It is here:http://arxiv.org/PS_cache/astro-ph/pdf/9803/9803097.pdfFLRW = Friedmann - Lemaitre Robertson Walker universe.Friedmann-Lemaitre is the common GR equation describing the dynamic evolution of a perfect fluid approx. to the universe. Roberston-Walker is the metric describing the spacetime of a homogeneous isotropic universe of this fluid. It is the common model used.Yes - when you are talking about inflation the expansion is very rapid and the energy for this is provided by the quantum field driving the expansion. Where this field comes (and exactly what form it takes)from is unknownNote that I answered you earlier in terms of a present epoch - i.e. applying standard GR solutions. Inflation is something altogether different.

Correct inflation is not the same as the current expansion.The dark energy is an unknown. There are quantum field models and of course the cosmological constant but we don't really know what it is.